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1. Denote by G the group of the equation upon which depends the determi. nation of the curves of order n 3 having simple contact at 1 n ( n -3 ) points with a given curve C of order n having no double points. The case in which n is odd was discussed in the former paper (Transactions, January, 1902) and G was shown to be a subgroup of the group defined by the invariants 43, 04, , , * *, the latter group being holoedrically isomorphic with the first hypoabelian grouip on 2p indices with coefficients taken modulo 2. For n even, G is contained in the group H defined by the invariants 04' 069 * with even subscripts. JORDAN has shown (Traite, pp. 229-242) that H is holoedrically isomorphic with the abelian linear group A on 2p indices with coefficients taken modulo 2. The object of the present paper is to establish the latter theorem by a short, elementary proof, which makes no use of the abstract substitutions [al, 1 ., p, p1] of JORDAN, and which exhibits explicitly the correspondence t between the substitutions of the isomorphic groups.

The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows

In the paper with the above title, which appeared in these T r a n s a c t i on s, vol. 2 (1901), p. 428, I obtained a number of general theorems concerning the zeros of functions of the form 02 y' ol y, where y is a solution of a hornogeneous linear differelntial equation of the second order. There s a further general theorem of the same character which I should have included in that paper if I had discovered it at the time, and which I now give as VII' below, it being a counterpart to VII in the earlier paper. I also give three special applications of this general result, numbered VIII', VIII", VIII"', since they are counterparts of VIII. A part of VIII' was obtained by Fite in the Annals of M athematics for June, 1917, (see the closing lines of his article) by another method; and it was this special result of Fite's that suggested to me the much more general Theorem VII'. What follows should be regarded as an addendum to the former paper, to be inserted on page 434 at the end of ? 2. All references are to that paper, to which the reader should turn for an explanation of the notation and a statement of the restrictions placed on the functions. VII'. If none of the six functions

In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together

It has been brought to my attention by Mr. N. E. Steenrod that the lemma of page 708 in the paper referred to is in error. The final assertion of the proof is false. It is therefore necessary to point out that the paper is not "disturbed" by this fault. For if one requires that the Pn, «= 1,2, • • •, of the lemma be arcs then the (altered) lemma does hold, since it is true that the connected sum of a perfect continuous curve and an arc is "perfect." One verifies that this restricted lemma is sufficient for the uses of the paper.

Walsh's Theorem II states that if the roots of f (z) are in the circular region C, and if z is exterior to C, then ai lies in C. This is precisely the result obtained by Laguerre as given on page 59 of volume I of his collected works, if expressed in non-homogeneous coordinates as on page 57. We shall use another form of Laguerre's Theorem (loc. cit., p. 57) to prove the underlying proposition on which Walsh bases his proof of his Theorem I, as follows: Every circle through any point z and its "derived point" a as defined by (1) either passes through all the roots off ( z ), or else has at least one root in the region interior to it, and at least one root in the exterior region. I have recently called attentiont to the fact that this is a corollary of a theorem of Bocher's on jacobians which has served as a starting point in Walsh's earlier papers. With a slight change in Walsh's notation, whereby we substitute z's for a's, the proposition from which Theorem I is deduced may be stated thus

It is the object of this paper to obtain theorems concerning the stability of the straight line solutions, and equilateral triangle solutions, respectively, in the problem of three bodies by means of the theorems and calculations of two preceding papers.t It is shown that the generalized theorems of Bohl can be applied to a neighborhood of the straight line solutions, with arbitrary masses, and to a neighborhood of the equilateral triangle solutions, if the masses are such that the characteristic exponents of the generating orbit are not all pure imaginaries. The mutual distances of the three masses are assumed constant on the generating orbit, in both cases.

This paper contains a number of disturbing misprints: Equation (2) p. 560 should read Gpf(x) = aoxPfm+ a1xPf11 + *** + amixpf + amx. Line 17 p. 561 read Ap(x) XBp(x) instead of Ap(x)Bp(x). The term perfect (volkommen) in Theorem 1 is used in the sense of Steinitz, Algebraische Theorie der Korper, edited by Hasse and Baer, pp. 50-51. Line 21 p. 562 should read Fp(x) = Qp(x) X (xP ax) + Ax. Equation (9) p. 562 should read A = aoa(Pm-l)/(P-1) + aia(P'1)/(P-1) + + am,2a0P+1 + am-,a + am. In the expression line 9 p. 563 the last term should be A(")x. Equation (17) p. 564 should read F'"(x) = Fnl(x)P -Fn-1(Wn)PFn-1(X). Line 8 from below p. 574 should read

This gives the desired result. It might be pointed out that Theorem 4 shows that Ilfn-gn I-->O In ?4 it is tacitly assumed that the measurable set E can be partitioned into measurable sets E,,. This is always the case in separable spaces. To proceed without this assumption it will not be necessary to assume that J is metric. The class So(E) is defined as the class of functions finitely valued on E. Such a function is one for which there is a decomposition of E into a finite number of disjoint measurable subsets on each of which it is constant. This basis necessitates only a slight rewording in a few places. In Lemma 1 the set E should be taken as a set in A. In Theorem 2 the words "functions uniformly continuous" should be replaced by "functions finitely valued." In the proof of Theorem 11 the sentence "Fix . . . continuous on e" should be worded "Fix e with 18(E-e)

Unfortunately, there is a mistake in the proof of Theorem 3.3 of our paper entitled Hypersurfaces with constant mean curvature in the complex space which appeared in the Transactions of the AMS (Vol. 339 (1993), 685-702). In the proof of this theorem we applied Hopf's maximum principle, which holds for a class of hypersurfaces satisfying an elliptic PDE, to a one-parameter family of hypersurfaces with constant mean curvature (cmc) of Q5 (using the notations of the paper) obtained by reflecting an initial cmc hypersurface of Q5 on a one-parameter family of totally geodesic hypersurfaces of Q5. However, while we know that the initial hypersurface satisfies an elliptic PDE since it is invariant by the S' group of isometries of Q5 and has cmc, the reflections of the hypersurface do not satisfy the equation since, although they have cmc, they are not S1 invariant any more. Therefore, Hopf's maximum principle cannot be used and the proof, as it stands in the paper, is not correct. This mistake was pointed out to us by Professor J. Eschenburg. So far, we have not found a way to correct it and, in fact, this seems to be a difficult question.

In this paper we prove that if G is a finitely generated pro-(finite nilpotent) group, then every subgroup Gn , generated by nth powers of elements of G, is closed in G. It is also obtained, as a consequence of the above proof, that if G is a nilpotent group generated by m elements xl, ... , xm, then there is a function f(m, n) such that if every word in x?l of length 1, we denote the subgroups of G generated by all nth powers an, a E G. A. Shalev conjectured that for any n the subgroup Gn is closed in G. This is the same as saying that for arbitrary integers m > 1, n > 1 there exists an integer N = N(m, n) such that in an arbitrary m-generated finite group G every product of nth powers of elements of G can be represented in the form a, **N where ai E G, 1 < i < N. Let us show, for example, that the existence of a function N(m, n) implies that Gn is closed. The subset M = {an . . *aN: al, ... , aN E G} of G is closed as the image of the compact G x ... x G under the continuous map (al, ., aN) -a an ... an Now we can consider the finite nilpotent group G/H, where H is an arbitrary open subgroup of G and so we have GnH = MH, which implies that Gn lies in the closure of M. Hence Gn = M. In this paper we prove Shalev's Conjecture when G is a pro-(finite nilpotent) Received by the editors February 8, 1994. 1991 Mathematics Subject Classification. Primary 20E18; Secondary 20F05. ? 1994 American Mathematical Society 0002-9947/94 $1.00 + S.25 per page

In this paper, using a homological theory of graded Lie algebras and the representation theory of A(') , we compute the root multiplicities of the hyperbolic Kac-Moody Lie algebra HA(') up to level 4 and deduce some interesting combinatorial identities. INTRODUCTION The hyperbolic Kac-Moody Lie algebras have been considered as the next natural objects of study after the affine case. While the affine Kac-Moody Lie algebras have been extensively studied for their close connections to areas such as combinatorics, modular forms, and mathematical physics, many basic questions regarding the hyperbolic case are still unresolved. For example, the behavior of the root multiplicities is not well understood. Feingold and Frenkel [F-F] and Kac, Moody, and Wakimoto [K-M-W] made some progress in this area. They computed the level 2 root multiplicities for the hyperbolic Kac-Moody Lie algebras HA(1) and HE(1), respectively. In [Kangl] and [Kang2], we introduced an inductive program to study the higher level root multiplicities and the principally specialized affine characters of a certain class of Lorentzian Kac-Moody Lie algebras. More precisely, we realize these Lorentzian Kac-Moody Lie algebras. as the minimal graded Lie algebras L = efEz Ln with local part V + Lo + V*, where Lo is an affine Kac-Moody Lie algebra, V is the basic representation of Lo, and V* is the contragredient of V. Thus L = G/I, where G = ~nEZ Gn is the maximal graded Lie algebra with local part V+Lo+ V* and I = eDnEZ I' is the maximal graded ideal of G which intersects the local part trivially. By developing a homological theory, we determined the structure of homogeneous subspaces Ln = G/II, as modules over the affine Kac-Moody Lie algebra Lo for certain higher levels. The Hochschild-Serre spectral sequences played an important role in determining the structure of I,n. For the hyperbolic Kac-Moody Lie algebra HA(1), we were able to determine the structure of L4 for n = 1, 2, ... , 5, and computed the principally specialized affine characters up to level 5. But the root multiplicities were computed up to level 3 only. In this paper, using the representation theory of A(1) developed Received by the editors June 4, 1990 and, in revised form, September 12, 1991. 1991 Mathematics Subject Classification. Primary 1 7B67, 1 7B65; Secondary 1 7B10. This work was supported by NSF grant DMS 8806371. () 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

The paper [I2] contains one false result, 2.6, and two others whose proofs require substantial repair: 2.5 and 2.24. All errors were discovered by one critical reader, Till Plewe. In 2.5 of [I2], which characterizes those sober spaces X that have a largest pointless sublocale pl(X), the last six words of proof are not true, e.g. in the real line with a generic point adjoined. Argue instead: The meet of {x}~ and pl+(^0 is dense in the irreducible space {x}~ . Now every dense sublocale of an irreducible space Y contains (i.e. D(Y) contains) the generic point y. For every sublocale of any locale is an intersection of complemented sublocales C whose complements are open n closed [I2], so it suffices to show that every such C dense in Y has y in it. Otherwise the complement C would contain y, so no closed subspace except Y contains C , so C is open; and as C is dense, C = 0, contradicting y e C . 2.6 says that for a dense-in-itself regular space X the locale pl(X) has the same weight as X. This is false—refuted by many pairs (X, X') of regular spaces with the same pi (pl(A^) « pl(X')) but different weights, e.g. the space Q of rationals and a subspace of sQ consisting of Q and one more point. The last result in the paper, 2.24, is that (with everything metrizable; in contrast to CVs) no nonzero pointless-absolute Fa locale exists. The correct proof, now presented, amounts to showing that ( 1 ) any nonzero Fa sublocale A of a pointless-absolute Os has a closed sublocale B = pl(C), C a Cantor set; that (2) B is not pointless-absolute Fa , being not Fa in a suitable metrizable extension pl(E) ; and (3) boosting the extension E of B to an extension of A . In [I2], (1) is done correctly in three lines, and four more lines of the proof (the fourth and the last three) do (2). For (3), a pushout construction is proposed; but it is not hard to check that the pushout need not be first countable. Instead use F. Hausdorffs theorem [H]

In 1977, the second author announced the following consistent neg- ative answer to a question of Katetov: Assuming MA+-GH , there is a compact nonmetric space X such that X1 is hereditarily normal. We give the details of this example, and construct another example assuming CH . We show that both examples can be constructed so that X2\A is perfectly normal. We also construct in ZFC a compact nonperfectly normal X such that X2\A is nor- mal. In his classical paper (K), Katetov showed that if X and Y are infinite compact spaces and X x Y is hereditarily normal, then X and Y are per- fectly normal. By Sneider's theorem that a compact space with a GVdiagonal is metrizable (S), Katetov concludes that if X is compact and X3 is heredi- tarily normal, then X is metrizable. He asked if the same conclusion could be obtained assuming only that X2 is hereditarily normal. In 1977, the second au- thor obtained a counterexample assuming Martin's Axiom plus the negation of the Continuum Hypothesis (MA + -CH). This result was announced in (Ny,), and an outline of the proof appeared there, although with many details omitted. A complete proof appears in this paper for the first time. We also construct a (necessarily different, as will be seen) counterexample assuming CH. Since any counterexample must be perfectly normal, it is probably not sur- prising that our examples are related to Alexandrov's double arrow space D = (0, 1) x {0, 1} with the lexicographic order topology, for it is in some sense the only known example in ZFC of a compact perfectly normal nonmetrizable space. (See (Gi) for a discussion of this.) The double arrow space has also been called the "split interval" because one can think of obtaining it by splitting each x € (0, 1) into two points x~ and x+ , and putting the order topology on these points, where x~ is declared to be less than x+ and otherwise the order is the natural one inherited by (0, 1). Now if A c (0, 1), let D(A) be the same as above but with only the points of A split; of course D(A) is just the quotient space of D obtained by identifying x~ with x+ for all x £ A. If A is uncountable, then D(A) is a compact perfectly normal nonmetrizable space.

In this paper we prove a Sobolev-Poincare inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Holder regularity of the weak solutions follows in a standard way. Let y = Z,7 ai(aij9 a ) be a second-order degenerate elliptic operator in divergence form with measurable coefficients. In this paper we shall obtain pointwise estimates for the weak solutions of Su = 0 (H61der continuity of the weak solutions and Harnack inequality for nonnegative solutions). Let us recall that the original results for elliptic operators were obtained by De Giorgi, Nash, and Moser. An extensive bibliography about the degenerate case can be found in [FLI, FL2, FS]. To introduce the results of the present paper, let us recall some recent results. In [FL1, FL2] a suitable metric d is associated with the differential operator Y in such a way that we obtain a new geometry which is natural for the degenerate operator as the Euclidean geometry is natural for the Laplace operator (or, more precisely, as a suitable Riemannian geometry is natural for a secondorder elliptic operator). In the smooth case, this idea is contained in many papers: we refer to [FP, NSW]. The basic results in [FL 1, FL2] are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the Hormander hypoellipticity condition for sum-of-squares operators. We note that the same idea is used in [NSW, S, J, V] to obtain pointwise estimates for sum-of-squares operators. On the other hand, a different class of degenerate elliptic operators is considered in [FKS]: instead of a geometrical degeneracy, a measure degeneracy is allowed. A typical example of this class is given by Yu = div(wo(x)Vu), where cl is a weight function belonging to the A2-class of Muckenhoupt. Unified results for a class containing both the operators in [FL 1] and in [FKS] have Received by the editors August 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E35, 35J70. Partially supported by G.N.A.F.A. of C.N.R. and M.U.R.S.T., Italy. ( 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

Let G G be a connected semisimple Lie group. If G G has finite center, Harish-Chandra used Eisenstein integrals to construct Schwartz class wave packets of matrix coefficients and showed that every K K -finite function in the Schwartz space is a finite sum of such wave packets. This paper is the second in a series which generalizes these results of Harish-Chandra to include the case that G G has infinite center. In this paper, the Plancherel theorem is used to decompose K K -compact Schwartz class functions (those with K K -types in a compact set) as finite sums of wave packets. A new feature of the infinite center case is that the individual wave packets occurring in the decomposition of a Schwartz class function need not be Schwartz class. These wave packets are studied to obtain necessary conditions for a wave packet of Eisenstein integrals to occur in the decomposition of a Schwartz class function. Applied to the case that f f itself is a single wave packet, the results of this paper yield a complete characterization of Schwartz class wave packets.

In this paper we study the homology of the moduli spaces of instantons associated to principal SU(n) bundles over the four-sphere. This is accomplished by exploiting an "iterated loop space" structure implicit in the disjoint union of all moduli spaces associated to a fixed SU(n) with arbitrary instanton number and relating these spaces to the known homology structure of the four-fold loop space on BSU(n) . Moduli spaces of instantons (self-dual connections with respect to a conformal class of metrics) associated to principal G-bundles Pk(G) = P over S4 have proven to be basic objects in modern geometry. Here G is any simple compact Lie group and k is the integer that classifies the bundle P and is referred to as the instanton number. We denote these moduli spaces by At (G). There is a natural inclusion (0. 1) ik (G): Z, (G) Qk BG induced by forgetting the self-duality condition where we have identified the moduli space of based gauge equivalence classes of all connections on P with the kth component of the four-fold loop space Q?4BG [5]. In a fundamental paper, Atiyah and Jones [5] studied the inclusion (0.1) for G = SU(2) (when S4 has its standard conformally flat metric) and posed several fundamental questions. In a series of papers Taubes (cf. [24-26]) proved several basic existence theorems, stability theorems in terms of k and provided a basis framework to describe how the topology of Ak changes as k increases. Taubes' work is much more general than we describe here in that he not only studied general Lie groups G but also replaced 54 by an arbitrary compact closed Riemannian four-manifold (with arbitrary conformal class of metric). In [7] it was observed that, over the four-sphere but with G = Sp(n), the disjoint union of .k over all k form a homotopy C4 space and that iterated loop space techniques may be profitably used to study H. (xk) for individual k. Certain computational results may be obtained immediately from these Received by the editors October 17, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 53C55, 58E05; Secondary 53C57, 55P35.

Although this paper circulated as a preprint for 12 years, an error in the argument for the Capelli identity went unnoticed until Professor M. Wakayama read the paper in order to review it. I thank him for his care and for bringing this error to my attention. The error occurs on page 566. The result at issue, that det II is central in '/(gyn), is true (and classical) and the general line of argument is sufficient to prove it, but the details of the computation are described incorrectly. The identity on line 3, page 566 is correct, but the description of the calculation drawn from it is not. Instead of terms of the form

In the first paper, Alexandrov’s Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as space-rigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz’s Theorem, and a related coordinatization of the proper realizations of a 3 3 -connected spherical polyhedron. Sample results show that: (a) a single k k -gonal block and a k k -gonal hole yield static rigidity if and only if the block and hole are k k -connected in a vertex sense; and (b) a 4 4 -connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.

The condition that all spherical diagrams in a 2-complex be reducible is shown to be equivalent to the condition that all finite branched covers be aspherical. This result is related to the study of equations over groups. Furthermore large classes of 2-complexes are shown to be diagrammatically reducible in the above sense; in particular, every 2-complex has a subdivision which admits a finite branched cover which is diagrammatically reducible. A consequence of the classical Riemann-Hurwitz formula for Riemann surfaces [6, p. 301] is the fact that a finite branched cover of an aspherical Riemann surface is also aspherical. However the property of asphericity of a 2-complex implying asphericity of finite branched covers fails for general 2-complexes. In fact we shall prove (Theorem 4.5 below) that this property holds precisely for those 2-complexes which are diagrammatically reducible. A 2-complex X is said to be diagrammatically reducible (abbreviated DR(X)) if for every combinatorial map f: S2 X of the 2-sphere into X (a "spherical diagram" in X) there is a pair of faces in the domain with an edge in common which are mapped mirrorwise by f across that edge. A closely related notion was introduced by Lyndon and Schupp [13] but this notion appears in its present form in a paper by Sieradski [14]. In that same paper Sieradski asked a question which would have implied that all classical knot complements in the 3-sphere are homotopy equivalent to diagrammatically reducible 2-complexes. We shall prove this latter assertion (Theorem 6.5) making use of our amalgamation theorem for diagrammatic reducibility (Theorem 5.4). The characterization of diagrammatic reducibility in terms of branched covers llows from a lifting theorem which may be of general interest. If f: X Y is a combinatorial map of combinatorial 2-complexes, then f induces a map of the link complexes of vertices Lf: Lx > Ly (the "star graphs" or "coinitial graph" in group theoretical language). The map f is said to be reduced if Lf is an immersion. We prove (Theorem 4.2) th t if f: X Y is a reduced map of 2-complexes with X finite, then there is a commutative diagram (see below) where qT: Z > Y is a finite branched cover and where g is injective off the zero skeleton of X. This result is Received by the editors July 9, 1986. 1980 MuthenluticsSulviectClussificwtios? (1985 Retvisios?). Primary20F05, 57M12; Secondary57M20. 1987 American Mathematical Scoclety 0002-9947/87 $1 .0() + $.25 pel page

Let L denote a simple Lie algebra over the complex numbers. In this paper, we classify and construct all simple L modules which may be infinite dimensional but have at least one 1-dimensional weight space. This completes the study begun earlier by the authors for the case of L = A,,. The approach presented here relies heavily on the results of Suren Fernando whose dissertation dealt with simple weight modules and their weight systems. 0. Introduction. Let L be a finite-dimensional simple Lie algebra over the complex field C having a Cartan subalgebra H and denote by C(L) the centralizer of H in the universal enveloping algebra U of L. If X: H -C is a weight function of a 1-dimensional weight space Mx in a simple L module M then q: C(L) -> C, defined by q(c)v = cv for v E Mx and c E C(L), is an algebra homomorphism called a mass function of M. Clearly X restricted to H is equal to X. Conversely, given any algebra homomorphism q: C(L) -C one can construct a unique simple L module which admits X as a mass function [4, 10]. In [4] the authors determined all algebra homomorphisms q: C(L) -C for the simple Lie algebras L of type An and using these classified all "pointed" An modules (where we call a module pointed if it is simple and has at least one 1-dimensional weight space). In this paper we complete the classification of all pointed L modules for arbitrary simple Lie algebras. The collection of all pointed L modules clearly includes the highest weight L modules and is included in the collection of all Harish-Chandra L modules relative to the Cartan subalgebra H (-i.e. simple L modules having a weight space decomposition and finite-dimensional weight spaces relative to H). The latter inclusion is strict since there exist examples of Harish-Chandra A2 modules in which every weight space is two dimensional. In the special case of A1 modules, every Harish-Chandra A1 module is pointed. Our approach to this problem makes heavy use of the results from Fernando's thesis [8]. In particular, from Fernando's results we see that in place of determining all algebra homomorphisms q: C(L) -C it suffices to find only those algebra homomorphisms X which are associated with so-called pointed "torsion free" L Received by the editors May 23, 1984. This paper was presented at the 1984 Canadian Mathematical Society's Summer Seminar on Lie algebras and related topics held at the University of Windsor. 1980 Mathematics Subject Classification. Primary 17B10.

An example of Chuang [1] shows that the main results of [2] are false as stated. The purpose of this note is to state the correct versions of these theorems. We shall use the notation in [2], and all references to results are from that paper. We begin by noting that all of the results in [2] before Theorem 4 are correct as stated, and that the correction needed in Theorem 4 requires a subsequent change in Theorem 7 and in Theorem 9. All other results in the paper are correct. The statement of Theorem 4 concerns a linear G*-DI f, all of whose exponents come from W, the ordered collection of k-tuples of outer derivations which are independent modulo the inner derivations. For any exponent w appearing in f and coming from W, let fW be the sum of all monomials in f with exponent w. The error in the proof of Theorem 4 is the assumption that if fw(x,y) is a G*-PI for R, then fw(xWnyw) is also an identify for R. This is true when no involution is present, or equivalently, when y does not appear in f. However, given an exponent w appearing in f, a relation between r and r* will not in general hold for rw and (r*)W, unless * commutes in End(R) with w. Thus the induction used in Theorem 4 fails. T-he most important feature of Theorem 4 can be salvaged, using essentially the proof given. For any w coming from (dl,...,dk) E W, let k be the length of w. If f E F is linear and has all its exponents coming from W, an exponent w appearing in f is said to be of longest length if no other exponent of f has longer length. The conclusion of Theorem 4 is correct for all exponents of longest length, and the following is what the statement of the theorem should be.

This paper is concerned with the quasi-linear evolution equation u'(t) + A(t, u(t))u(t) = 0 in [0, T], u(0) = xo in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a "limit solution." The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed. This paper is concerned with the quasi-linear evolution equation u'(t) + A(t, u(t))u(t) = 0 in[O, T], u(O) = xo in a Banach space setting. The spirit of this inquiry follows that of T. Kato. Kato wrote a fundamental paper on linear evolution equations in 1953 [9]; that is, investigation of u'(t) + A(t)u(t) = 0 on[O, T], u(0) = x0. He strengthened and extended his analysis of the linear problem in 1970 [11]. Kato also wrote on the quasi-linear problem in 1975 [13]. We feel that our results give a natural approach to dealing with the quasi-linear problem. After discussing the setting and method of attack, our theorem is stated and proved. We then give an application of the theorem using the Sobolevskii-Tanabe theory of linear evolution equations of parabolic type. A proposition relevant to our theorem is also given. Let X and Y be Banach spaces, with Y densely and continuously embedded in X. Let x0 E Y, T > 0, r > r, > 0, r2 > 0, W= Bx(xo; r), Z = Bx(xo; rl) n By(xo; r2), and for each t E [0, T] and w E W, let -A (t, w) be the infinitesimal generator of a strongly continuous semigroup of bounded linear operators in X, with Y c D(A(t, w)). We consider the quasi-linear evolution equation v'(t) + A(t, v(t))v(t) = 0. (QL) Received by the editors December 14, 1978 and, in revised form, July 6, 1979. AMS (MOS) subject classifications (1970). Primary 34G05, 47D05; Secondary 65J05, 41A65.

Let A A be a commutative unital Banach algebra. Suppose G ⊂ A G \subset A is such that | | a | | ⩽ | | g a | | ||a|| \leqslant ||ga|| for all g ∈ G , a ∈ A g \in G,a \in A . Two questions are considered in the paper. Does there exist a superalgebra B B of A A in which every g ∈ G g \in G is invertible? Can one always have also | | g − 1 | | ⩽ 1 ||{g^{ - 1}}|| \leqslant 1 if g ∈ G g \in G ? Arens proved that if G = { g } G = \{ g\} then there is an algebra containing g − 1 {g^{ - 1}} , with | | g − 1 | | ⩽ 1 ||{g^{ - 1}}|| \leqslant 1 . In the paper it is shown that if G G is countable B B exists, but if G G is uncountable, this is not necessarily so. The answer to the second question is negative even if G G consists of only two elements.

Let L and K be differential fields with L an extension of K. It is shown how the module of Kahler differentials S41/K can be used to "linearize" properties of a differential field extension L/K. This is done without restriction on the characteristic p and yields a theory which for p # 0 is no harder than the case p = 0. As an application a new proof of the Ritt basis theorem is given. Introduction. Let k be a field and A a finitely generated algebra over k. One can study the properties of A by taking a surjective homomorphism of k-algebras 4: k[y1, ... ,y.] -> A where Yi ... , y,n are indeterminates over k and then analyzing the kernel I of 4. This is for instance done in [1] for the case of A an integral domain. There one assumes that 4 is chosen so that 0(Yi), ..., 4)(yp) is a transcendance basis for the quotient field of A over k. One then chooses nonzero Fj in k[y1,...,yj] nfI and dj in N for p < j < n such that (1) dj is the degree of Fj in yj, (2) Degreeyi Fj < di if p < i < j, (3) dj is as small as possible. One then observes that if Sj is defined by Fj = Sjy4i + terms of degree less than dj inyj and if S = Sp+, ... Sn, then S 4 I and k[y1, .... ,Yn, I/S]I is generated as an ideal of k[y, . .. ,Yn, 1/S] by p+1,~ .. * * Fn. The subject of this paper is the study of a finitely generated differential algebra over a differential field. This matter has of course been considered before. However to prove anything beyond the most trivial statements one has customarily generalized the above-mentioned procedure, and this has led ultimately to a fairly complicated theory. Such theory is more complicated in nonzero characteristic than in characteristic zero. The purpose of this paper is to show that we may achieve our results in another way, and so that procedure is not followed here. Instead differential algebras are studied by reducing the number of derivation operators involved one at a time. To illustrate this method I have included a new proof of Seidenberg's version of the Ritt-Raudenbusch basis theorem. The primary tool used in reducing the number of derivation operators is the theory of Kiihler differentials, a method for "linearizing" much of the ring theory Received by the editors June 14, 1972 and, in revised form, December 15, 1972. AMS (MOS) subject classifications (1970). Primary 12H05.

In this paper we study a class of kernels FR which generalize the Bochner-Riesz kernels on the N-dimensional torus. Our main result consists in upper estimates for the LP norms of FR as R tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces. 1. Throughout this paper we identify the N-dimensional torus TN (N > 2) with the N-dimensional cube QN = {x E R N: 1/2 0 we form the means FR* g(t) = f(R -m)g(m)exp(2'7imt) (1) m where t E TN, m ranges on the integer lattice ZN, and * denotes the convolution in TN. In several cases one is able to estimate the L 1( TN) norm of the kernel FR(t) = E f(R 1m)exp(2'srimt) (2) m and (if 0 E S and f(O) = 1) deduce convergence results for the means (1) (as R -x oc) when g belongs to certain classes of periodic functions (see e.g. [1]-[3], [7], [8], [12]). For instance, if f(x) = (1 -I X12)8 when lxl 1, then FR = K,A is the familiar Bochner-Riesz kernel. In this case K. I. Babenko (cf. [1]) has shown that IIKRIIl is of the same order as R(N-1)/2 as R tends to infinity (N > 2, 0 < 8 < (N 1)/2), while Stein proved the estimate IIKR Il log R if 8 = (N 1)/2 [9]. Recently Yudin [12] obtained the estimate lF = O(R (N1)/2) if f is the characteristic function of a closed balanced set C, whose boundary has finite upper Minkowski measure. The method of Yudin (which also yields estimates for the LP( TN) norms of FR) uses Jackson approximation theorem, which in tum involves estimates for the L2(R N) modulus of continuity of f. Such a method can be adapted to a more general situation, as we show in this paper. In ??2-4 we consider kernels of the type (2) associated with functionsf whose derivatives of certain orders are controlled by (not necessarily positive) Received by the editors May 13, 1980. 1980 Mathematics Subject Classification. Primary 42B99; Secondary 46E35.

The first-order theory of arborescent structures is shown to be completely faithfully interpretable in the first-order theory of trees. It follows from this interpretation that Vaught's conjecture is true for arborescent structures, the theory of arborescent structures is decidable, and every MK0-categorical arborescent structure has a decidable theory. Arborescent structures were introduced in [8] by abstracting a very simple property that trees possess. The main result of [8] is that every consistent, recursively axiomatizable theory of arborescent structures has a recursive model. In this paper we continue the study of arborescent structures, showing as the principal result that the theory of arborescent structures is interpretable in the theory of trees in a very strong way. From this, many interesting results are deducible; for example, the theory of arborescent structures is decidable (Corollary 4.2) and Vaught's conjecture is true for arborescent structures (Corollary 4.4). For a better appreciation of the significance of the notion of arborescence, some familiarity with the development leading to its isolation might be useful. Rabin [3] proved the decidability of S2S, and from this result the decidability of (more than) the first-order theory of trees easily follows by interpreting it in S2S. My interest in the theory of trees originated after proving in [5] that every p0-categorical theory of trees is decidable. This result was later shown to be connected with that of Rabin's when I proved in [4] that if a theory is interpretable in S2S, then so is every one of its Ho-categorical completions. At this point I became interested in finding some rather simple conditions which implied interpretability in S2S, or in the theory of trees, possibly some sort of structural property. I first considered partially ordered sets, and was able to extend the above results for trees to reticles [7], which are partially ordered sets not embedding the 4-element partially ordered set by showing that the theory of reticles is interpretable in the theory of trees. Reticles turned out to be a fascinating kind of partially ordered set which appeared to possess many of the nicer properties that trees do. I tried for some time to modify the interpretation of the theory of reticles in the theory of trees in order to Received by the editors June 4, 1980. 1980 Mathematics Subject Classification. Primary 03C65, 03C35. 'Research supported in part by NSF Grant MCS-7905028. Much of this paper was completed while I was visiting the University of California, San Diego, during the year 1978-1979. ? 1981 American Mathematical Society 0002-9947/81 /0000-0368/$04.75

By using the quadratic form and unbounded operator theory a new approach to the general dilation theory is presented. The boundedness condition is explained in terms of the Friedrichs extension of symmetric operators. Unbounded dilations are introduced and discussed. Applications are given to various problems involving positive definite functions. t. Introduction. There are two principal conditions in general bounded dilation theory: positive definiteness and the boundedness condition. While the first one is naturally justified and generally accepted, the second one is rather complicated, usually not easy to verify, and several simplifications of this condition are known under some additional assumptions. This paper originated as an attempt to explain and understand this boundedness condition. It occurred to us that the positive quadratic form theory provides an appropriate framework for such.an explanation. However, consequences of this theory are much deeper than just an explanation of the boundedness condition. They lead naturally to a new general dilation theory, which deals not only with bounded but also with unbounded dilations. In §2 of this paper some known and some new results on positive quadratic forms and unbounded operators are discussed. In §3 a general dilation theory is presented, which contains the known bounded dilation theory as a special case. §4 deals with several applications of the previous results, in particular to *-semigroups, *-algebras, Gramians, operator moment problems, and quantum mechanics. The general reference to quadratic forms and unbounded operators used here is [8]. There is an extensive literature on the bounded dilation theory. A general treatment can be found in [6, 7, t]. All linear spaces are assumed to be complex. If X, Y are linear spaces, then L(X, Y) (B(X, Y), respectively) stands for the linear space of all linear (bounded linear, if X, Y are normed, respectively) mappings from X to Y. Moreover, L(X) = L(X,X), B(X) = B(X,X). The semigroup structure in L(X) or B(X) is always the multiplicative one, with the composition. Ix or I denotes the identity operator in X. A subspace of a Hilbert space H is a linear subset of H. An operator T in H is a linear mapping from a subspace D(T) of H into H. D(T), kerT denote the domain and the kernel of T, respectively, C°°(T) denotes the intersection of all D(Tn), n = 1,2,.... Let H,K be Hilbert spaces, let D(T) be a subspace of Received by the editors May 15, 1986. 1980 Mathematics Subject Cklssficatvm. Primary 47D05, 43A35; Secondary 47B25, 81E05, 47A20.

We construct a (mod-I) Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a (mod-I) Spanier-Whitehead dual. Introduction. In this paper we establish the existence of a (mod-i) SpanierWhitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. This generalises the familiar construction of the Spanier-Whitehead dual for a compact complex manifold. In the forthcoming papers [J-2 and J-3] we make use of this to establish a Becker-Gottlieb type transfer for proper and smooth maps of smooth quasi-projective varieties. Recall that associated to every finite spectrum X there exists another spectrum, denoted DX, and called the Spanier-Whitehead dual of X, which is characterised by the following property. Let EO denote the sphere spectrum, while E denotes any arbitrary spectrum. Then there exists a map ,u: E* X A DX of spectra which induces isomorphisms [u]: h-q(X, E) -* hq(DX, E) and [T,p]: h-q(DX, E) -* hq(X, E) for all q. Here r is the map interchanging the two factors X and DX while h* ( , E) (h* ( , E)) is the generalised homology (generalised cohomology, respectively) with respect to the spectrum E. (See [Sw, pp. 321-335] for a general reference on the familiar notion of Spanier-Whitehead duality in topology.) If X happens to be the suspension spectrum associated to a compact closed real manifold M, there exists an explicit geometric construction of its SpanierWhitehead dual. If a is the normal bundle to imbedding M in a large sphere as a smooth closed submanifold, then a suitable desuspension of the Thom space of this bundle forms a Spanier-Whitehead dual for M+. We observe that this construction therefore provides a Spanier-Whitehead dual for any compact complex manifold, by merely forgetting its complex structure. This construction is generalised here for projective and geometrically unibranched varieties over any algebraically closed field. Received by the editors November 14, 1984 and, in revised form, June 23, 1985. 1980 Mathematics Subject Classification. Primary 14F35; Secondary 14F20, 55P25, 55P42, 55N20. (@)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

0. Introduction. In this paper we consider homologically trivial, locally smooth actions of finite and compact Lie groups on topological 4-manifolds having the same homology as CP2, the complex projective plane. Our main result states that all such actions are lowally complex linear and that the acting groups can also act linearly on CP2. In particular, we obtain the classification of groups acting on both manifolds homotopy equivalent to CP2: G acts on the complex projective plane (resp. on the Chern manifold) if and only if G is isomorphic to a subgroup (resp. a pseudofree subgroup) of PU(3). The paper is organized as follows. In §1 a precise statement of the main result is given. §2 examines tangent representations of some cyclic groups; the main tool being the G-signature theorem. The relevant number theoretic computations are carried out in Appendix 1. §§3 and 4 discuss abelian and nonabelian groups in terms of the numbers of fixed points (Euler characteristic of the fixed point set). §5 proves that all tangent representations are complex. §§6-9 discuss several classes of groups acting without fixed points. The terminology that we use to distinguish among different classes of fixed point free groups is adopted from the linear case. §10 contains the proof of our main theorem and, finally, §11 is devoted to group actions on the Chern manifold. For completeness, a list of fixed point free linear groups is included in Appendix 2. ACKNOWLEDGMENT. I would like to thank Professor John Ewing for suggesting this problem to me and also for his support and encouragement. I have been informed that I. Hambleton and R. Lee have independently obtained results similar to those presented here.